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What is a statistical model?
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This paper addresses two closely related questions, "What is a statistical model?" and "What is a parameter?" The notions that a model must "make sense," and that a parameter must "have a well-defined meaning" are deeply ingrained in applied statistical work, reasonably well understood at an instinctive level, but absent from most formal theories of modelling and inference. In this paper, these concepts are defined in algebraic terms, using morphisms, functors and natural transformations. It is argued that inference on the basis of a model is not possible unless the model admits a natural extension that includes the domain for which inference is required. For example, prediction requires that the domain include all future units, subjects or time points. Although it is usually not made explicit, every sensible statistical model admits such an extension. Examples are given to show why such an extension is necessary and why a formal theory is required. In the definition of a subparameter, it is shown that certain parameter functions are natural and others are not. Inference is meaningful only for natural parameters. This distinction has important consequences for the construction of prior distributions and also helps to resolve a controversy concerning the Box-Cox model.
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“We argue that, as suggested in [46], it has a role to play in the study of point cloud data as well, and we give two illustrations of how this could happen, within the context of clustering.”
From Topology and data (2009) · cites this paper as [46]
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Cites
Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2003) · doi:10.1201/9780429258480 · Cited by 27,276
(2005) · Springer eBooks · doi:10.1007/0-387-27255-0_9 · Cited by 20,333
McCullagh, P., & Nelder, J. A. (1989) · doi:10.1007/978-1-4899-3242-6 · Cited by 16,836
Self-citationAn Analysis of Transformations
Box, G. E. P., & Cox, D. R. (1964) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1964.tb00553.x · Cited by 15,356
Jeffreys, H., Moran, P. A. P., & Chatfield, C. (1922) · Nature · doi:10.1038/109132a0 · Cited by 7,877
Statistical Decision Theory and Bayesian Analysis
Berger, J. O. (1985) · Springer Series in Statistics · doi:10.1007/978-1-4757-4286-2 · Cited by 7,722
Nelder, J. A., & Wedderburn, R. W. M. (1972) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2344614 · Cited by 7,703
Spatial Interaction and the Statistical Analysis of Lattice Systems
Besag, J. (1974) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1974.tb00999.x · Cited by 6,067
Statistics and Causal Inference
Holland, P. W. (1986) · Journal of the American Statistical Association · doi:10.1080/01621459.1986.10478354 · Cited by 5,293
Regression Models for Ordinal Data
McCullagh, P. (1980) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1980.tb01109.x · Cited by 4,453
Self-citation
Research works printed but not linked: 15
- Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
- MCCULLAGH, P. (2000). Invariance and factorial models (with discussion). J. Roy. Statist. Soc. Ser. B 62 209-256. · DOI
- GELMAN, A., CARLIN, J. B., STERN, H. and RUBIN, D. B. (1995). Bayesian Data Analy sis. Chapman and Hall, London. · DOI · Paper page
- MAC LANE, S. (1998). Categories for the Working Mathematician, 2nd ed. Springer, New York.
- MCCULLAGH, P. (1999). Quotient spaces and statistical models. Canad. J. Statist. 27 447-456. · DOI
- WICHURA, M. (2001). Some de Finetti ty pe theorems. Preprint.
- DE FINETTI, B. (1975). Theory of Probability 2. Wiley, New York.
- HELLAND, I. S. (1999a). Quantum mechanics from sy mmetry and statistical modelling. Internat. J. Theoret. Phy s. 38 1851-1881. · DOI
- MCCULLAGH, P. (1980). Regression models for ordinal data (with discussion). J. Roy. Statist. Soc. Ser. B 42 109-142. · DOI · Paper page
- TJUR, T. (2000). Contribution to the discussion of McCullagh (2000). J. Roy. Statist. Soc. Ser. B 62 238-239. · DOI
- BESAG, J. E. (1974). Spatial interaction and the statistical analysis of lattice sy stems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192-236. · DOI · Paper page
- BESAG, J. E. and HIGDON, D. M. (1999). Bayesian analysis of agricultural field experiments (with discussion). J. Roy. Statist. Soc. Ser. B 61 691-746. · DOI · Paper page
- BESAG, J. E. and KOOPERBERG, C. L. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733-746. · DOI · Paper page
- FRASER, D. A. S. (1968b). The Structure of Inference. Wiley, New York.
- MACCULLAGH, J. (1839). An essay towards the dy namical theory of cry stalline reflexion and refraction. Trans. Roy. Irish Academy 21 17-50.
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BackgroundGaussian Random Field Models for Spatial Data
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BackgroundScaling Multidimensional Inference for Structured Gaussian Processes
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“Whereas the subsequent works of McCullagh (2002), Besag (2002), Besag and Mondal (2005) and McCullagh and Clifford (2006) have provided new perspectives on spatial modelling via continuum random fields, Lee and Nelder (1996, 2001) and Lee et al. (2006) revived Henderson’s (1950, 1975) h-likelihood…”
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“[6, 8, 9, 10, 11, 18, 19, 23, 27]).”
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Gómez, L., Ospina, R., & Frery, A. C. (2019) · Remote Sensing · doi:10.3390/rs11040385 · Cited by 13
“Albeit an expressive estimator, the distribution of M is not known and, therefore, cannot be used as a statistical model [42] for prediction and testing [43,44].”
MethodWhen should modes of inference disagree? Some simple but challenging examples
Fraser, D. A. S., Reid, N. M., & Lin, W. (2018) · The Annals of Applied Statistics · doi:10.1214/18-aoas1160sf · Cited by 12
Lew, M. J. (2019) · Handbook of experimental pharmacology · doi:10.1007/164_2019_286 · Cited by 13
“tion of errors in his observations, and about facts that are assumed to be known for the purposes of statistical calculations. |[Colquhoun, 1971, p. v] Statistical models can take a variety of forms [McCullagh, 2002], but the model for the familiar Student’s t-test for independent samples is reasonably representative.…”
BackgroundSensitivity of Regular EstimatorsPreprint
Mukhin, Y. (2018) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.1805.08883 · Cited by 12
Revisiting the Large n (Sample Size) Problem: How to Avert Spurious Significance Results
Spanos, A. (2023) · Stats · doi:10.3390/stats6040081 · Cited by 11
González, J., & von Davier, M. (2013) · Journal of Educational Measurement · doi:10.1111/jedm.12018 · Cited by 11
Crane, H. (2015) · Bernoulli · doi:10.3150/14-bej617 · Cited by 11
A frequentist framework of inductive reasoningPreprint
Bickel, D. R. (2016) · Cited by 11
Noventa, S., Ye, S., Kelava, A., & Spoto, A. (2024) · Psychometrika · doi:10.1007/s11336-024-09950-z · Cited by 10
“A statistical model is identifiable if no two different sets of parameters values are observation-ally equivalent, i.e., they yield the same likelihood of the observed outcomes (see, e.g., Bamber and Van Santen, 2000; McCullagh, 2002, for a detailed introduction to the topic).”
BackgroundLinear Regression with Optimal Rotation
Livadiotis, G. (2019) · Stats · doi:10.3390/stats2040028 · Cited by 10
Manisera, M., & Zuccolotto, P. (2015) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2015.05.011 · Cited by 10
Multi-Party Inference and Uncongeniality
Meng, X. (2011) · International Encyclopedia of Statistical Science · doi:10.1007/978-3-642-04898-2_381 · Cited by 10
General Fitting Methods Based on Lq Norms and their Optimization
Livadiotis, G. (2020) · Stats · doi:10.3390/stats3010002 · Cited by 9
Metadata
- Authors
- McCullagh, P.
- Affiliations
- University of Chicago (US)
- Corresponding author
- McCullagh, Peter
- ORCID
- McCullagh, Peter
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2002-10-01
- Volume
- 30
- Issue
- 5
- Pages
- 1225-1310
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2049954094
- Cited by
- 248
- References
- 103
- Citations by year
- 2026: 9 · 2025: 22 · 2024: 13 · 2023: 5 · 2022: 10 · 2021: 23 · 2020: 17 · 2019: 16 · 2018: 18 · 2017: 12 · 2016: 12 · 2015: 16 · 2014: 12 · 2013: 10 · 2012: 7
- FWCI
- 5.37
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- CBS Research Portal (Copenhagen Business School) (submittedVersion)
- Funders
- National Science Foundation (0071726, 9705347)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- statistical models, parameters, natural parameters, morphisms, functors, natural transformation, model extension, prediction, prior distributions
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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